Research Proposal on Arithmetic Geometry

柴教授将继续他在主极化阿贝尔簇的模空间中线性亚簇的性质的研究。他还将考虑PEL类型的Shimura品种减产的精细结构。这是他最近关于PEL型Shimura变种的普通Hecke轨道密度的结果的自然推广。在一个新的研究方向上，柴教授将尝试将L上同调的技巧应用于最近图论工作中出现的特殊指数和。这项工作是在数论的现代领域-算术代数几何中进行的。数论是研究1、2、3等计数的学科，是数学中最古老的分支。顾名思义，算术代数几何是三个不同数学领域的综合体。在过去的五十年里，数学家们发现，几何中熟悉的对象，比如看起来像多项式方程的图形的曲线，可以通过将它们视为抽象的代数结构来更好地理解。这个强大的想法使几何学发生了革命性的变化。随着时间的推移，数学家们意识到，他们的许多最困难的问题都可以提出，然后通过这些几何对象的抽象代数描述来解决。正如数学中经常发生的那样，这种不断增长的抽象水平实际上产生了许多非常具体和实际的结果。

Ching-Li Chai9800609Professor Chai will continue his study of the properties of linear subvarieties in the moduli space of principally polarized abelian varieties. He will also consider the fine structure of the reduction of the Shimura varieties of PEL-type,. This is a natural extension of his recent results on the density of ordinary Hecke orbits of Shimura varieties of PEL-type. In a new direction of investigation, Professor Chai will try to apply the technique of l-adic cohomology to particular exponential sums that have arisen in recent work in graph theory.This work in a modern area of Number Theory known as arithmetic algebraic geometry. Number Theory is the study of the counting numbers, 1, 2, 3, etc. and is the oldest branch of mathematics. Just as the complicated name suggests, arithmetic algebraic geometry is a synthesis of three different areas of mathematics. In the last fifty years, mathematicians have found that familiar objects from geometry, like the curves that appear as the graph of a polynomial equation, can be much better understood by viewing them as abstract algebraic constructions. This powerful idea revolutionized geometry. In time number theorists realized that many of their most difficult problems could be posed and then solved in terms of these abstract algebraic descriptions of geometric objects. As often happens in mathematics, this ever increasing level of abstraction has actually produced many very concrete and practical results.